Outside of Star Trek, Anglim is best known for his portrayal of the deformed John Merrick in the original stage production of The Elephant Man, both of which co-starred Kevin Conway. For his performance of Merrick in the Broadway production of the play, Anglim was nominated for a 1979 Best Actor Tony Award. The play was also taped and aired on television in 1982, which led to his receiving Emmy Award and Golden Globe nominations.

Anglim graduated from Yale University with a BA in English literature in 1973; that same year, he made his film debut with an uncredited bit part in the boxing drama The All-American Boy. Following a stint with the Southbury Playhouse in Connecticut, Anglim appeared in the 1976 PBS mini-series The Adams Chronicles, which also featured Richard Cox in the cast. Anglim then had a role in the controversial film The Message (a depiction of the founding of Islam), which starred Michael Ansara and Michael Forest.

Following the success of The Elephant Man, Anglim continued performing on Broadway, playing the title role in a revival of William Shakespeare's MacBeth. This production also starred Ivar Brogger, Kelsey Grammer and John Vickery and was filmed for a television broadcast later that year. Among the actors to replace Anglim in The Elephant Man until the play closed in 1981 were Bruce Davison, Jeffrey Hayenga, and the actor best known for playing Luke Skywalker, Mark Hamill. Additionally, Concetta Tomei became a cast member late in the production.

Anglim also had a supporting role in the 1983 film Testament and starred with Alice Krige in the 1988 drama film Haunted Summer. In the meantime, Anglim continued performing on the stage. In 1989, he co-starred with Mark Harelik and Dendrie Taylor in Howard Korder's Search and Destroy. The following year, he worked with Brian Reddy in M. Butterfly.

After a few more films in the early 1990s (The Man Inside in 1990 and Milena in 1991), Anglim stuck solely to television. Besides appearing multiple times on DS9, he also guest-starred in an episode of Millennium (working with Megan Gallagher, Terry O'Quinn and Bill Smitrovich) and appeared in the 1998 Dallas TV special War of the Ewings along with Tracy Scoggins. This was his most recent television appearance to date.